Reduced Loop Quantum Gravity with Scalar fields Jurekfest Warsaw - PowerPoint PPT Presentation

Reduced Loop Quantum Gravity with Scalar fields Jurekfest Warsaw University 16-20.09.2019 Kristina Giesel, Institute for Quantum Gravity, FAU Erlangen

Marseille 2004 Loops and Spinfoams May 3rd to 7th 2004 CPT Luminy, Marseille

Dynamics in Loop Quantum Gravity Start with classical general relativity Ashtekar-Barbero variables In canonical approach: Apply canonical quantization End up with a quantum version of Einstein’s classical equations: Quantum Einstein Equations We can use either Dirac or reduced quantization In both approaches quantum dynamics crucially depends on choices on makes in step of quantization Different models exists for dynamics: Physical properties? Associated Spin foam models [Kieselowksi, Lewandowski ‚19]

Reduced Quantization: LQG Three tasks to perform: 1.) Derive physical phase space: Construct Dirac observables for GR 2.) Derive gauge invariant version of Einstein‘s equations on physical phase space: Determine physical Hamiltonian 3.) Quantize reduced system: Quantum Einstein Equations on H Phys

Relational Formalism: Observables [Rovelli, Dittrich] ( q A , P A ) , { C I } , I labelset Start with constrained theory Choose for each constraint a so called reference fields (clock) { T I } s . t . { T I , C J } ≈ δ I J Then given phase space function f, associated observable is: ∞ 1 X O C n !( τ I − T I ) n { f, C I } ( n ) f,T ( τ ) = Matter clocks n =0 Choose clocks { O f , C I } ≈ 0 Dirac observables: from matter dof { O f , O g } ≈ O { f,g } ∗ Algebra of observables: dO f,T Gauge invariant dynamics = { O f , H phys } on reduced phase space: d τ

Ide Which Reference Matter? Introduce additional scalar fields coupled to gravity Distinguish between 2 classes of models: Type I and type II models [K.G., T. Thiemann `12] Alternatively one can choose geometrical dof as reference field: 'geometrical clocks’ —> quantization more complicated Geometrical clocks have been considered in the context of linear cosmological perturbation theory [K.G., Herzog`17, K.G., Herzog, Singh ’18, K.G., Singh, Winnekens ’19]

Ide Reference Matter: ( T I , ρ , W j ) Lagrangian can obtain up to 8 scalar fields: L D = � 1 p | g | ( g µ ν [ ρ r µ T 0 r ν T 0 + α ( ρ ) V µ V ν + 2 β ( ρ )( r µ T 0 ) V ν ] + Λ ( ρ )) 2 V µ := W j r µ T j α ( ρ ) , β ( ρ ) , Λ ( ρ ) arbitrary functions of ρ with T µ ν matter can be interpreted as dust, has same Particular models considered so far: L T D : α = β = Λ = ρ [Brown, Kuchar `95] L ND : α = 1 , β = Λ = ρ = 0 [Bicak, Kuchar `97] L NRD : α = β = 0 , Λ = ρ [Brown, Kuchar `95] L GD : α = 0 , β = 1 , Λ = ρ [Kuchar, Torre `91]

Ide Canonical Analysis of Type I & II ρ 6 = 0 We distinguish between cases: β ( ρ ) 6 = 0 (II) α ( ρ ) = β ( ρ ) = 0 α ( ρ ) 6 = 0 (I) or In both cases one obtains system with 2nd class constraints ρ , W j are determined by solving 2nd class constraints strongly We end up with: (I) ( A, E ) , ( T 0 , P 0 ) , ( T j , P j ) 4 additional dof (II) ( A, E ) , ( T 0 , P 0 ) 1 additional dof Particular cases: (I) L T D : h 0 ( A, E, T 0 ) h 0 ( A, E ) all other models depends on h 0 ( A, E, T 0 ) ∂ a T 0 q ab ∂ a T 0 ∂ b T 0 (II) h 0 ( A, E, T 0 ) depends only on h 0 ( A, E ) BK-M

Ide Example: Type II Lagrangian obtains 1 scalar field: T 0 I := � 1 p 2 g µ ν ( r µ T 0 )( r ν T 0 ) L S = | g | L ( I ) , Particular models considered so far: I := � 1 p 2 g µ ν ( ⇥ µ T 0 )( ⇥ ν T 0 ) L S = | g | L ( I ) , Klein-Gordon field: [Rovelli, Smolin `93] [Kuchar, Romano`95] General case: [Thiemann `06] In both cases constraints are of the form: e e C a = P 0 T 0 ,a + C geo C 0 = P 0 + h 0 ( A, E ) = 0 ( A, E ) a

Ide Summary: Reference Matter Type I models: (i) Reduction wrt to Diffeo and Hamilton in classical theory Type II models: Reduction wrt Hamilton in classical theory, Diffeo via Dirac quantization in quantum theory Difference relevant once quantization is considered

Ide Beautiful Beaches KITP Workshop Santa Barbara: Fishbowl @ KITP: Beach next to KITP:

Ide Reduced Dynamics We have derived (partially) reduced phase space of GR O A,T I ( τ , σ k ) , O E,T I ( τ , σ k ) ( O A,T 0 ( τ ) , O E,T 0 ( τ )) can be interpreted as physical time parameter Aim: Gauge invariant version of Einstein‘s eqn: d d τ O A,T I ( τ , σ k ) = { O A,T I ( τ , σ k ) , H phys } d d τ O E,T I ( τ , σ k ) = { O E,T I ( τ , σ k ) , H hys } Question: How does look like for different models?

Ide Reduced Dynamics One can show that for all considered models: Type I: (i) Z d 3 σ H ( σ ) H phys = Type I: (ii) S Type II: Particular Models: q ( C geo ) 2 − q ab C geo C geo L T D H ( σ ) = a b L NRD H ( σ ) = | C geo | ( σ ) L GD H ( σ ) = C geo ( σ ) r q −√ qC geo + √ q L S ( C geo ) 2 − q ab C geo C geo H ( σ ) = a b

Reduced Quantization: LQG What kind of current models exist for LQG? [K.G., T. Thiemann `12] Type I models: 4 scalar fields Examples: Brown-Kuchar dust model [K.G., T. Thiemann `07] Gaussian dust model [K.G., T. Thiemann `12] 4 KG scalar field [K.G., Vetter `16, K.G., Vetter ’19] Type II models: Partial reduction, only reference field associated with Hamiltonian constraint Examples: 1 KG scalar field [Domagala, K.G., Kaminski, Lewandowski `10] 1 Gaussian dust field [Husein, Pawlowski `11]

Ide Two scalar field Models In this talk we focus on two particular models Type II: One massless Klein-Gordon scalar field Refer to as 'Warsaw model’, Dirac quantization Type I: Four massless Klein-Gordon scalar fields Refer to as '4 scalar fields model’, Reduced Quantization Allows comparison of different models and in particular allows first steps of comparison between Dirac and reduced quantization Both can be seen as generalizations of the APS model to full LQG [Ashtekar, Pawlowski, Singh 2006]

Ide Warsaw Model: Reference Matter Idea: Use one scalar field to reduce wrt Hamiltonian constraint ✓ √ gR − 1 ◆ Z √ gg µ ν ϕ ,µ ϕ , ν d 4 X S = 2 Diffeos are solved at the quantum level, Quantum Dirac observables Reference field is one massless scalar field [Ashtekar, Pawlowski, Singh 2006] In order to formulate the model we need: H di ff diffeomorphism invariant Hilbert space H di ff geometric operators on to construct quantum Dirac observables ˆ H phys H di ff on (a suitable domain of)